Model efficiency is an important area of model management,

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2 Roll Your Own By Bob Crompton Model efficiency is an important area of model management, and model compression is one of the dimensions of such efficiency. Model compression improves efficiency by creating a significantly reduced number of model points compared to a seriatim model. Cluster modeling is a model compression methodology that has been successfully implemented for a number of years. A good introduction to cluster modeling can be found in the article Cluster Analysis: A Spatial Approach to Actuarial Modeling. 1 For simplicity, this article is referred to as the Milliman article. Some actuarial software incorporates cluster modeling; if yours doesn t, this article is for you. SOFTWARE USED The clustering in this article is performed with the open source software R. 2 In addition to the software included in the standard installation, I used two additional packages xlsx and fpc to perform some of the tasks discussed in this article. To install these packages, use the R console commands: install.packages( xlsx, dependencies = TRUE) install.packages( fpc, dependencies = TRUE) Since these packages are not part of the home library, they will need to be added. They can be manually loaded as follows: library(xlsx) library(fpc) CREATING A CLUSTER MODEL I obtained from a colleague an in-force file of universal life (UL) policies. There are five plan types in the file and 1,347 records. The broad steps we need to create a cluster model are: Generate synthetic policy attributes. Apply weighting to the attributes as appropriate. Split data into segments. Import the file of in-force attributes to R. Apply the clustering algorithm. Export results. Reconfigure in-force files for the reduced number of cells and rerun the model. These steps are considered in this article. GENERATE SYNTHETIC POLICY ATTRIBUTES One of the most interesting aspects of the Milliman article is the use of synthetic policy attributes that is, policy attributes that are not found either in the in-force file or are not simple transformations of data found in the in-force file. Quinquennial ages are examples of simple transformations of in-force data. The development of clusters uses attributes associated with projected cash flows in addition to the attributes found in the in-force file. For example, the life/health model used in the Milliman article includes the following synthetic attributes: Present value of proxy profits Present value of proxy profits through 10 projection years Present value of proxy profits through 20 projection years There are three other attributes used in this model, of which two are synthetic. The only native attribute is beginning reserve. It is instructive to review the synthetic attributes for the term life model included in the Milliman article: Beginning reserve Cumulative present value of proxy cash flows Present value of proxy cash flows Years 1 5 Years 6 10 Years Years Years Years Projected death benefits Years 1 5 Years 6 10 Years Years Years Years Projected premiums Years 1 5 Years 6 10 Years Years Years Years DECEMBER 2016 THE MODELING PLATFORM

3 This is 20 attributes, of which only one is native, with all the rest being synthetic. Why are there so many? The complexity of reserving for level term, combined with the mortality patterns during and after the level term period, mean extensive information is required if we want a well-fitting cluster model. The attributes I used for my model were those used in the Milliman article for the traditional life/health model with one exception. I did not include the present value of total proxy profits because my original projections only went for 20 years. The information contained in the early years proxy profits and the later years proxy profits overlaps the information contained in total proxy profits. APPLY WEIGHTS TO ATTRIBUTES Weights adjust the relative importance of the various attributes used for clustering. Weighting affects both the selection of the representative cell for a cluster as well as the cells assigned to the cluster. Consistent with the cluster attributes, the weights I used for my model were those used in the Milliman article for the traditional life/health model. CREATE SEGMENTS The in-force file needs to be split into segments. The Milliman article describes segments as follows: You divide the business into segments, which instructs the program not to map across segment boundaries. Segments might include plan code, issue year, GAAP [generally accepted accounting principles] era or any other dimension of interest. So clustering is applied at the segment level. In my example, the entire file is treated as one segment. If I had used multiple segments, they would have been based on plan code. IMPORTING THE DATA INTO R For the import operation, I am demonstrating how to import from Excel because Excel is ubiquitous. I will demonstrate two possibilities because there is more than one way to skin a cat. (I don t actually know this from personal experience, but generations of folk wisdom attest to the truth of this statement. Who am I to question generations of folk wisdom?) Importing Directly from Excel For the first import operation, I use the read.xlsx function. The read.xlsx function is from the xlsx package. Use the following console command: MyInforce <- read.xlsx( c:/inforce.xlsx, 1) # 2nd parameter indicates which tab to import DECEMBER 2016 THE MODELING PLATFORM 15

4 Roll Your Own MyInforce, the variable containing the output from the read.xlsx operation, is a data frame. Data frames are formatted matrix-like data structures. They are convenient since they can be used in many built-in functions that require matrix input. Importing From a Comma-Separated File For this import operation, I use the read.table function: MyInforce <- read.table( c:/inforce.csv, header = TRUE, sep =, ) The read.table operation yields a data frame, the same as the read.xlsx operation. Comments on Excel vs. CSV Files The read.xlsx function has the obvious benefit of convenience. You are working directly from Excel so you don t have to reformat. In addition, if you put each segment in a separate tab, it is easy to loop through the tabs with the read.xlsx function. DIY clustering is an easy and straightforward process using existing code. However, there are some serious drawbacks to using the read. xlsx approach: It s s-l-o-w! An Excel file with 10,000 records took about 35 minutes to load using read.xlsx. It s memory intensive. My computer was unable to load a file of 25,000 records because the Java back-end to xlsx ran out of memory. Unless all of your segments are less than 2,000 or so records, reading directly from Excel may not be in the cards. Note that the R manual on data import/export has the following warning: The most common R data import/export question seems to be how do I read an Excel spreadsheet. The first piece of advice is to avoid doing so if possible! If you have access to Excel, export the data you want from Excel in tab-delimited or comma-separated form, and use read.delim or read.csv to import it into R. 3 However, if you are committed to using Excel as your data source, the manual contains a description of a few other R packages that provide Excel import capability. Experiment with some or all of these to see if they work any better than the xlsx package. CSV files don t have either of these problems. I was able to load a file of 100,000 records in less than 5 seconds using the read.table function. The only issue I have noted with CSV files is that if you forget to reformat comma-separated numeric values, chaos and darkness will result. APPLY CLUSTERING ALGORITHM Once the data is in R, it is simple to create clusters. There are a number of cluster functions available. I used the pam function because the output contains the information we need to create the clusters. Pam is based on a version of the K-means approach to clustering. The following code is for the cluster model with 13 cells. Note that the data is standardized by setting one of the function parameters shown. Standardization often gives better results when using clustering algorithms. fit <- pam(myinforce, 13, stand = TRUE) The output variable fit is a list containing, among other things, the index number of the representative cells for each cluster in the component id.med, and the cluster assignment of each of the inforce records in the component clustering. So the component id.med is a vector with length equal to the number of clusters (in this instance, 13), and the component clustering is a vector with length equal to the number of in-force records. Table 1 Comparison of Policy Attributes ($1,000s) Attribute Original Clustered Ratio Initial reserve 522, , % Projected first-year premiums 77, , % Projected first-year benefits 38, , % Present value proxy profits, years , , % Present value proxy profits, years (104,775.4) (109,964.9) 105.0% 16 DECEMBER 2016 THE MODELING PLATFORM

5 We pass these back from R using the following export code: write.table(fit$id.med, c:/cluster_index.csv, sep =, ) write.table(fit$id.clustering, c:/cluster_ assignment.csv, sep =, ) It is possible to use the write.xlsx approach to write directly to Excel files; however, the same caveats regarding speed and memory mentioned in the discussion of importing the data will apply to exporting the data as well. RECONFIGURING THE IN-FORCE FILE Once the cluster assignments and the cluster representative cells are determined, reconfiguring the in-force file is straightforward. Static items such as issue age, sex, plan code and similar items are set equal to these items from the representative cell. Dynamic items such as initial reserve, initial fund, amount of insurance and similar items are summed over all the in-force records belonging to the cluster. RESULTS OF THE TEST FILES The results from the test clustering are shown in Table 1. I compare the clustered vs. unclustered results for the total net cash flows and the totals of the synthetic attributes. Observations on Fit Given that this model has a 99 percent compression ratio (13 cells compared to 1,347 in the original model), the fit is reasonable. Model projections for premiums, benefits and distributable earnings are shown in Figure 1. Although the purpose of this article is merely to show how to create cluster models, a brief discussion of how to improve the fit might be helpful. There are two obvious possibilities. The first is to adjust the weighting factors. For example, the weightings for both early and late proxy profits could be increased. This would likely improve the fit for first-year benefits as well as for proxy profits. The second obvious adjustment is to split the model into three or four segments, rather than just one segment. Three segments with four cells each might perform better at fitting the original data since the different UL plans have differing patterns of cash flows and profit emergence. OTHER THINGS TO CONSIDER In addition to simply generating and identifying clusters, there are several other things we can consider as we create cluster models. Figure 1 Model Projections for Premiums, Benefits and Distributable Earnings Original Model Premiums 90,000,000 80,000,000 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000, ,000,000 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 10,000, ,000,000 30,000,000 25,000,000 20,000,000 15,000,000 10,000,000 5,000,000 0 Distributable Earnings Original Model Original Model Optimal Number of Clusters What is the optimal number of clusters? If we don t care much about model fit, and are mainly concerned about processing time, then obviously one cluster is optimal. In that case, we simply define the cluster based on the average policy number. ( This is technically known as humor. It is not intended to be taken seriously.) The function pamk in the package fpc estimates the optimal number of clusters based on optimum average silhouette width. A cluster silhouette is a measure of how close each point in one cluster is to points in the neighboring clusters. The further away the points are, the better. DECEMBER 2016 THE MODELING PLATFORM 17

6 Roll Your Own Interestingly, using the pamk function to investigate the optimal number of clusters in the range of two to 100 results in an optimal cluster count of three. That seems to explain how the 99 percent compression model performed as well as it did. Of course, silhouette optimization is not what actuaries are really interested in, so this result does not mean we should automatically choose three clusters. Testing for Sensitivity to Order of Attributes Many K-means algorithms used for clustering seem to be based on the expectation-maximization algorithm. There are some anecdotes that these algorithms are sensitive to the order in which the attributes are presented. The documentation for the pam function claims it is a more robust version of K-means. 4 It is not clear if this means it does not exhibit sensitivity to the order of attributes. I did not bother tracing back to the source reference for the algorithm, but it is something users can easily test. I tested for this sensitivity by running pam with two additional input files that differed only in the column order of the data. Both additional files produced clusters identical to the original in-force file. CONCLUSION As presented here, DIY clustering is an easy and straightforward process using existing code. As my father-in-law used to tell me, It s easy once you know how. In R, once you know how, many things are insanely easy. The difficult part of R is that it is so extensive, finding the right package to do just what you want is a time-consuming task. ENDNOTES Bob Crompton, FSA, MAAA, is a vice president of Actuarial Resources Corporation of Georgia, located in Alpharetta, Ga. He can be reached at bob.crompton@arcga.com. 1 Avi Freedman and Craig Reynolds, Cluster Analysis: A Spatial Approach to Actuarial Modeling, Milliman Research Report, August 2008, uploadedfiles/insight/research/life-rr/cluster-analysis-a-spatial-rr pdf. 2 R Core Team, R: A Language and Environment for Statistical Computing (Vienna: R Foundation for Statistical Computing, 2016), 3 This can be accessed at 4 The pam function is contained in the package cluster: Maechler, M., Rousseeuw, P., Struyf, A., Hubert, M., Hornik, K. (2015). cluster: Cluster Analysis Basics and Extensions. R package version The quickest way to access the documentation for pam is to enter??pam at the command prompt in R. You can also access the documentation at 18 DECEMBER 2016 THE MODELING PLATFORM